Elastodynamic fields near running cracks by finite elements

A finite element method is developed for the computation of elastodynamic stress intensity factors at a rapidly moving crack tip. The method is restricted to bodies whose surfaces and two-material interfaces are either parallel to the direction of propagation or are sufficiently remote. The crack tip starts to move at the instant that it is struck by an incident wave. The finite element grid moves undeformed with the crack tip. The main result consists in the fact that the method of non-singular calibrated crack tip elements, in which the stress-intensity factor is determined from its statically calibrated ratio to the crack opening displacement in an adjacent node, is extended to dynamic problems with moving cracks, for both in-plane and anti-plane motions. The dependence of the calibration ratio on the crack tip velocity is established from previously developed analytical solutions for the near-tip displacement fields. Numerical results compare favorably with known analytical solutions for cracks moving in an infinite solid. The grid motion causes an apparent asymmetric additional damping matrix.     

Large-Strain Generalization of Microplane Model for Concrete and Application

The formulation of the microplane model for concrete and development of model M4 in the three preceding companion papers in this study is here extended to large strains. After giving examples of certain difficulties with the second Piola-Kirchhoff stress tensor in the modeling of strength and frictional limits on weak planes within the material, the back-rotated Cauchy (true) tensor is introduced as the stress measure. The strain tensor conjugate to the back-rotated Cauchy (or Kirchhoff) stress tensor is unsuitable because it is nonholonomic (i.e., path-dependent) and because its microplane components do not characterize meaningful deformation measures. Therefore Green's Lagrangian tensor is adopted, even though it is not conjugate. Only for this strain measure do the microplane components of the strain tensor suffice to characterize the normal stretch and shear angle on that microplane. Using such nonconjugate strain and stress tensors is admissible because, for concrete, the elastic parts of strains as well as the total volumetric strains are always small, and because the algorithm used guarantees the energy dissipation by large inelastic strains to be nonnegative. Examples of dynamic structural analysis are given.     

Confinement-Shear Lattice Model for Concrete Damage in Tension and Compression: I. Theory

The mechanical behavior of the mesostructure of concrete is simulated by a three-dimensional lattice connecting the centers of aggregate particles. The model can describe not only tensile cracking and continuous fracture but also the nonlinear uniaxial, biaxial, and triaxial response in compression, including the postpeak softening and strain localization. The particle centers representing the lattice nodes are generated randomly, according to the given grain size distribution, and Delaunay triangulation is used to determine the lattice connections and their effective cross-section areas. The deformations are characterized by the displacement and rotation vectors at the centers of the particles (lattice nodes). The lattice struts connecting the particles transmit not only axial forces but also shear forces, with the shear stiffness exhibiting friction and cohesion. The connection stiffness in tension and shear also depends on the transversal confining stress. The transmission of shear forces between particles is effected without postulating any flexural resistance of the struts. The shear transmission and the confinement sensitivity of lattice connections are the most distinctive features greatly enhancing the modeling capability. The interfacial transition zone of the matrix (cement mortar or paste) is assumed to act approximately in series coupling with the bulk of the matrix. The formulation of a numerical algorithm, verification by test data, and parameter calibration are postponed for the subsequent companion paper.     

A Computational Treatment of 35 IPR Isomers of C88

Abstract

The complete sets of 35 isolated-pentagon-rule (IPR) isomers of C₈₈ is described by the SAM1 (Semi-Ab-initio Model 1) quantum-chemical method. The separation energetics is also computed at the HF/STO-3G, HF/3-21G, and HF/4-31G levels. The SAM1 and HF/4-31G data mostly agree within a few kJ/mol. As the SAM1 energetics does not reproduce the recent NMR observations, entropy contributions are included, too, being based on the harmonic-oscillator and rigid-rotator model. Considerable temperature effects on the relative stabilities in the system are found. The ground-state structure of C₈₈ is a C ₃ isomer, however, with an increase of temperature a C ₂ structure becomes important. At still higher temperatures a near C ₂ species is dominant. The results can be viewed as a good agreement with the available observations, and they further expand the family of the IPR sets where the thermodynamic equilibrium treatment allows for a satisfactory support of observations.     

Computer simulation of evolving capillary bridges in granular media

A numerical method for the simulation of spatially evolving liquid–vapour interfaces in arbitrary two dimensional granular media is presented. Solid- and liquid-phase objects are described by polynomials whose edges evolve according to surface tension forces until a prescribed equilibrium contact angle at three-phase contact points and a constant mean curvature on two-phase contact lines is achieved. The main advantage of the method is the possibility to account for topological transitions (interface coalescence or rupture) and direct calculation of the force acting on solid interfaces due to liquid bridges. The method has been validated by comparing numerical and analytical results for a single pendular liquid bridge and then demonstrated on the simulation of transition from the pendular to funicular and capillary state in a wet particle assembly.     

Bifurcation and stability of structures with interacting propagating cracks

A general method to calculate the tangential stiffness matrix of a structure with a system of interacting propagating cracks is presented. With the help of this matrix, the conditions of bifurcation, stability of state and stability of post-bifurcation path are formulated and the need to distinguish between stability of state and stability path is emphasized. The formulation is applied to symmetric bodies with interacting cracks and to a halfspace with parallel equidistant cooling cracks or shrinkage cracks. As examples, specimens with two interacting crack tips are solved numerically. It is found that in all the specimens that exhibit a softening load-displacement diagram and have a constant fracture toughness, the response path corresponding to symmetric propagation of both cracks is unstable and the propagation tends to localize into a single crack tip. This is also true for hardening response if the fracture toughness increases as described by an R-curve. For hardening response and constant fracture toughness, on the other hand, the response path with both cracks propagating symmetrically is stable up to a certain critical crack length, after which snapback occurs. A system of parallel cooling cracks in a halfspace is found to exhibit a bifurcation similar to that in plastic column buckling.     

Practical prediction of creep and shrinkage of high strength concrete

Model for practical prediction of creep and shrinkage of normal strength concrete, developed previously, is extended to high strength concrete. It is found that only a minor adjustment for the concrete strength effect is needed in the formulas for drying creep. The formulas for basic creep and shrinkage need no adjustment. The prediction model is compared with test data for creep and shrinkage obtained recently by Ngab, Nilson and Slate, and by Collepardi, Corradi and Valente, and a satisfactory agreement is demonstrated. The coefficient of variation of the deviations from test data is not larger than that for the normal cient of variation of the deviations from test data is not larger than that for the normal strength range. However, the existing data are rather limited and further testing is desirable.     

Three-dimensional harmonic functions near termination or intersection of gradient singularity lines: A general numerical method

The harmonic function u near point 0 from which a single singularity ray emanates is assumed to be dominated by the term r^λρ^ρU where r = distance from point 0, p = known constant and ρ = chosen function of angular spherical coordinates θ, ?, for which a partial differential equation with boundary conditions, especially those at the singularity rays, and a variational principle, are derived. Because grad U is nonsingular, a 1numerical solution is possible, using, e.g. the finite difference or finite element methods. This reduces the problem to finding λ of the smallest real part satisfying the equation Det (A_ij) = 0 where A_ij is a large matrix whose coefficients depend linearly on μ = λ(λ + 1). In general λ and A_ij are complex. Solutions can be obtained either by reduction to a standard matrix eigenvalue problem for μ, or by successive conversions to nonhomogeneous linear equation systems. Computer studies have confirmed the feasibility of the method and have shown that highly accurate results can be obtained. Solutions for cracks and notches ending at a plane or conical surface, and for cracks ending obliquely at a halfspace surface, are presented. In these cases, λ is real and the singularity is always weaker (λ > p) than on the singularity line and may even disappear (λ > 1). Furthermore, elastic stresses under a wedge-shaped rigid sliding stamp or at a corner of a crack edge, and also harmonic functions at three-sided pyramidal notches, have been analyzed. Here λ < p was found to occur. A simple analytical solution for one class of special cases has also been found and used to check some of the numerical results.     

Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions

The fatigue crack path has been studied on a tensile specimen with holes. The experimental crack path trajectories were compared with those calculated numerically. To incorporate the influence of constraint on the crack curving, we predicted the fatigue crack path by using the two-parameter modification of the maximum tensile stress (MTS) criterion. The values of the mixed-mode stress intensity factors K_I, and K_II as well as the corresponding constraint level characterized by T-stress were calculated for the obtained curvilinear and reference crack path trajectories. It is shown that in the studied configuration the effect of T-stress on the crack path is not significant. On the other hand the effect of constraint on the fatigue crack propagation rate is more pronounced.